Extreme points of a class of subordinate functions
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- by T. Sheil-Small PDF
- Proc. Amer. Math. Soc. 91 (1984), 73-74 Request permission
Abstract:
It is shown that, if $F\left ( z \right )$ is subordinate to $H\left ( {\left ( {1 + z} \right ) / \left ( {1 - z} \right )} \right )$ in the unit disc, where $H\left ( w \right )$ is a quadratic polynomial for which $H’\left ( w \right ) \ne 0$ in the right half-plane, then \[ F\left ( z \right ) = \int \limits _T {H\left ( {\frac {{1 + z{e^{ - it}}}}{{1 - z{e^{ - it}}}}} \right )d\mu } \] for a suitable probability measure $\mu$ on the unit circle $T$.References
- David A. Brannan and James G. Clunie (eds.), Aspects of contemporary complex analysis, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 623462
- D. J. Hallenbeck and T. H. MacGregor, Support points of families of analytic functions described by subordination, Trans. Amer. Math. Soc. 278 (1983), no. 2, 523–546. MR 701509, DOI 10.1090/S0002-9947-1983-0701509-8
- I. S. Jack, Functions starlike and convex of order $\alpha$, J. London Math. Soc. (2) 3 (1971), 469–474. MR 281897, DOI 10.1112/jlms/s2-3.3.469
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 73-74
- MSC: Primary 30C80
- DOI: https://doi.org/10.1090/S0002-9939-1984-0735567-8
- MathSciNet review: 735567